Hostname: page-component-f554764f5-c4bhq Total loading time: 0 Render date: 2025-04-19T06:50:31.967Z Has data issue: false hasContentIssue false
  • English
  • Français

Phase-Field Models for Multi-Component Fluid Flows

Published online by Cambridge University Press:  20 August 2015

Junseok Kim
Junseok Kim*
Affiliation:
Department of Mathematics, Korea University, Seoul 136-701, Republic of Korea
*
*Corresponding author.Email:[email protected]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper, we review the recent development of phase-field models and their numerical methods for multi-component fluid flows with interfacial phenomena. The models consist of a Navier-Stokes system coupled with a multi-component Cahn-Hilliard system through a phase-field dependent surface tension force, variable density and viscosity, and the advection term. The classical infinitely thin boundary of separation between two immiscible fluids is replaced by a transition region of a small but finite width, across which the composition of the mixture changes continuously. A constant level set of the phase-field is used to capture the interface between two immiscible fluids. Phase-field methods are capable of computing topological changes such as splitting and merging, and thus have been applied successfully to multi-component fluid flows involving large interface deformations. Practical applications are provided to illustrate the usefulness of using a phase-field method. Computational results of various experiments show the accuracy and effectiveness of phase-field models.

Keywords

76D0576D4576T3082C26Navier-StokesCahn-Hilliardmulti-componentsurface tensioninterface dynamicsinterface capturingphase-field model
Type
Research Article
Information
Communications in Computational Physics , Volume 12 , Issue 3 , September 2012 , pp. 613 - 661
Copyright
Copyright © Global Science Press Limited 2012

References

[1]Acar, R., Simulation of interface dynamics: a diffuse-interface model, Visual Comput., 25 (2009), 101–115.Google Scholar
[2]Almgren, A. S., Bell, J. B., Colella, P., Howell, L. H. and Welcome, M. L., A conservative adaptive projection method for the variable density incompressible Navier-Stokes equations, J. Comput. Phys., 142 (1998), 1–46.CrossRefGoogle Scholar
[3]Ames, W. F., Numerical Methods for Partial Differential Equations, Academic Press, San Diego, 1992.Google Scholar
[4]Anderson, D. M., McFadden, G. B. and Wheeler, A. A., Diffuse-interface methods in fluid mechanics, Ann. Rev. Fluid Mech., 30 (1998), 139–165.Google Scholar
[5]Badalassi, V. E. and Banerjee, S., Nano-structure computation with coupled momentum phase ordering kinetics models, Nucl. Eng. Des., 235(10-12) (2005), 2005–1107.Google Scholar
[6]Badalassi, V. E., Ceniceros, H. D. and Banerjee, S., Computation of multiphase systems with phase field models, J. Comput. Phys., 190 (2003), 371–397.CrossRefGoogle Scholar
[7]Barrett, J. W., Blowey, J. F. and Garcke, H., Finite element approximation of the Cahn-Hilliard equation with degenerate mobility, SIAM J. Numer. Anal., 37 (1999), 286–318.CrossRefGoogle Scholar
[8]Barrett, J. W., Blowey, J. F. and Garcke, H., On fully practical finite element approximations of degenerate Cahn-Hilliard systems, M2AN Math. Model. Numer. Anal., 35 (2002), 713–748.CrossRefGoogle Scholar
[9]Bell, J., Collela, P. and Glaz, H., Second-order projection method for the incompressible Navier-Stokes equations, J. Comput. Phys., 85 (1989), 257–283.CrossRefGoogle Scholar
[10]Berger, M. J. and Rigoustsos, I., Technical Report NYU-501, New York University-CIMS, 1991.Google Scholar
[11]Blowey, J. F., Copetti, M. I. M. and Elliott, C. M., Numerical analysis of a model for phase separation of a multi-component alloy, IMA J. Numer. Anal., 16 (1996), 111–139.Google Scholar
[12]Borcia, R. and Bestehorn, M., Phase-field for Marangoni convection in liquid-gas systems with a deformable interface, Phys. Rev. E, 67 (2003), 066307.Google Scholar
[13]Boyer, F., A theoretical and numerical model for the study of incompressible mixture flows, Comput. Fluids, 31(1) (2002), 2002–41.CrossRefGoogle Scholar
[14]Boyer, F. and Lapuerta, C., Study of a three component Cahn-Hilliard flow model, M2AN, 40(4) (2006), 2006–653.Google Scholar
[15]Boyer, F., Lapuerta, C., Minjeaud, S., Piar, B. and Quintard, M., Cahn-Hilliard/Navier-Stokes model for the simulation of three-phase flows, Transp. Porous Media, 2009.CrossRefGoogle Scholar
[16]Brackbill, J. U., Kothe, D. B. and Zemach, C., A continuum method for modeling surface tension, J. Comput. Phys., 100 (1992), 335–354.CrossRefGoogle Scholar
[17]Caffarelli, L. A. and Muler, N. E., An L ∞ bound for solutions of the Cahn-Hilliard equation, Arch. Rational Mech. Anal., 133 (1995), 129–144.Google Scholar
[18]Cahn, J. W., Free energy of a nonuniform system II: thermodynamic basis, J. Chem. Phys., 30 (1959), 1121–1124.Google Scholar
[19]Cahn, J. W., On spinodal decomposition, Acta Metall., 9 (1961), 795–801.CrossRefGoogle Scholar
[20]Cahn, J. W. and Hilliard, J. E., Free energy of a non-uniform system I: interfacial free energy, J. Chem. Phys., 28 (1958), 258–267.CrossRefGoogle Scholar
[21]Cahn, J. W. and Hilliard, J. E., Free energy of a nonuniform system III: nucleation in a two-component incompressible fluid, J. Chem. Phys., 31 (1959), 688–699.Google Scholar
[22]Ceniceros, H. D., Nos, R. L. and Roma, A. M., Three-dimensional, fully adaptive simulations of phase-field fluid models, J. Comput. Phys., 229 (2010), 6135–6155.Google Scholar
[23]Chacha, M., Radeev, S., Tadrist, L. and Occelli, R., Numerical treatment of the instability and breakupof a liquid capillary column in a bounded immiscible phase, Int. J. Multiphase Flow, 23 (1997), 377–395.Google Scholar
[24]Chang, Y. C., Hou, T. Y., Merriman, B. and Osher, S., A level set formulation of Eulerian interface capturing methods for incompressible fluid flows, J.Comput. Phys., 124 (1996), 449–464.Google Scholar
[25]Chella, R. and Viñals, J., Mixing of a two-phase fluid by cavity flow, Phys. Rev. E, 53 (1996), 3832–3840.CrossRefGoogle ScholarPubMed
[26]Chen, L. Q. and Shen, J., Applications of semi-implicit Fourier-spectral method to phase field equations, Comput. Phys. Commun., 108 (1998), 147–158.Google Scholar
[27]Chorin, A. J., A numerical method for solving incompressible viscous flow problems, J. Comput. Phys., 2 (1967), 12–26.Google Scholar
[28]Dehghan, M., Finite difference procedures for solving a problem arising in modeling and design of certain optoelectronic devices, Math. Comput. Simul., 71 (2006), 16–30.Google Scholar
[29]Ding, H. and Spelt, P. D. M., Wetting condition in diffuse interface simulations of contact line motion, Phys. Rev. E, 75 (2007), 046708.Google Scholar
[30]Ding, H., Spelt, P. D. M. and Shu, C., Diffuse interface model for incompressible two-phase flows with large density ratios, J. Comput. Phys., 226 (2007), 2078–2095.CrossRefGoogle Scholar
[31]Elliott, C. M. and French, D. A., Numerical studies of the Cahn-Hilliard equation for phase separation, IMA J. Appl. Math., 38 (1987), 97–128.CrossRefGoogle Scholar
[32]Eyre, D. J., www.math.utah.edu/∼eyre/research/methods/stable.ps.Google Scholar
[33]Eyre, D. J., Computational and Mathematical Models of Microstructural Evolution, The Material Research Society, Warrendale, 1998.Google Scholar
[34]Fernandino, M. and Dorao, C. A., The least squares spectral element method for the Cahn-Hilliard equation, Appl. Math. Model., 35 (2011), 797–806.CrossRefGoogle Scholar
[35]Fife, P. C., Models for phase separation and their mathematics, Euro. J. Diff. Eqns., 48 (2000), 1–26.Google Scholar
[36]Furihata, D., A stable and conservative finite difference scheme for the Cahn-Hilliard Equation, Numer. Math., 87 (2001), 675–699.Google Scholar
[37]Furihata, D., Onda, T. and Mori, M., A finite difference scheme for the Cahn-Hilliard equation based on a Lyapunov functional, GAKUTO Int. Series Math. Sci. Appl., 2 (1993), 347–358.Google Scholar
[38]Garcke, H., Nestler, B. and Stoth, B., On anisotropic order parameter models for multi-phase systems and their sharp interface limits, Phys. D, 115 (1998), 87–108.CrossRefGoogle Scholar
[39]Gilbarg, D. and Trudinger, N. S., Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer-Verlag, Berlin, 2001.CrossRefGoogle Scholar
[40]Glimm, J., Grove, J. W., Li, X. L., Shyue, K. M., Zhang, Q. and Zeng, Y., Three-dimensional front tracking, SIAM J. Sci. Comput., 19 (1998), 703–727.Google Scholar
[41]Gomeza, H. and Hughes, T. J. R., Provably unconditionally stable, second-order time-accurate, mixed variational methods for phase-field models, J. Comput. Phys., 230 (2011), 5310–5327.Google Scholar
[42]Gueyffier, D., Li, J., Nadim, A., Scardovelli, R. and Zaleski, S., Volume-of-fluid interface tracking with smoothed surface stress methods for three-dimensional flows, J. Comput. Phys., 152 (1999), 423–456.Google Scholar
[43]Gurtin, M. E., Polignone, D. and Viñals, J., Two-phase binary fluids and immiscible fluids described by an order parameter, Math. Models Meth. Appl. Sci., 6 (1996), 815–831.Google Scholar
[44]Harlow, F. H. and Welch, J. E., The MAC method: a computing technique for solving viscous, incompressible, transient fluid flow problems involving free surface, Phys. Fluids, 8 (1965), 2182–2189.Google Scholar
[45]He, Q., Glowinski, R. and Wang, X. P., A least-squares/finite element method for the numerical solution of the Navier-Stokes-Cahn-Hilliard system modeling the motion of the contact line, J. Comput. Phys., 230 (2011), 4991–5009.Google Scholar
[46]He, Q. and Kasagi, N., Phase-field simulation of small capillary-number two-phase flow in a microtube, Fluid Dyn. Res., 40(7-8) (2008), 2008–497.CrossRefGoogle Scholar
[47]Jacqmin, D., Calculation of two-phase Navier-Stokes flows using phase-field modeling, J. Comput. Phys., 155 (1999), 96–127.CrossRefGoogle Scholar
[48]Jacqmin, D., Contact-line dynamics of a diffuse fluid interface, J. Fluid Mech., 402 (2000), 57–88.Google Scholar
[49]Kan, H. C., Shyy, W., Udaykumar, H. S., Vigneron, P. and Tran-Son-Tay, R., Effects of nucleus on leukocyte recovery, Ann. Biomed. Eng., 27 (1999), 648–655.CrossRefGoogle ScholarPubMed
[50]Kan, H. C., Udaykumar, H. S., Shyy, W. and Tran-Son-Tay, R., Hydrodynamics of a compound drop with application to leukocyte modeling, Phys. Fluids, 10 (1998), 760–774.Google Scholar
[51]Karniadakis, G. E., Israeli, M. and Orszag, S. A., High-order splitting methods for the incompressible Navier-Stokes equations, J. Comput. Phys., 97 (1991), 414–443.Google Scholar
[52]Keestra, B. J., Puyvelde, P. C. J. V., Anderson, P. D. and Meijer, H. E. H., Diffuse interface modeling of the morphology and rheology of immiscible polymer blends, Phys. Fluids, 15(9) (2003), 2003–2567.CrossRefGoogle Scholar
[53]Khatavkar, V. V., Anderson, P. D., Duineveld, P. C. and Meijer, H. H. E., Diffuse interface modeling of droplet impact on a pre-patterned solid surface, Macromol. Rapid. Commun., 26 (2005), 298–303.CrossRefGoogle Scholar
[54]Khatavkar, V. V., Anderson, P. D., Duineveld, P. C. and Meijer, H. H. E., Diffuse-interface modelling of droplet impact, J. Fluid Mech., 581 (2007), 97–127.CrossRefGoogle Scholar
[55]Khatavkar, V. V., Anderson, P. D. and Meijer, H. H. E., Capillary spreading of a droplet in the partially wetting regime using a diffuse-interface model, J. Fluid Mech., 572 (2007), 367–387.Google Scholar
[56]Kim, J. S., A continuous surface tension force formulation for diffuse-interface models, J. Comput. Phys., 204(2) (2005), 2005–784.Google Scholar
[57]Kim, J. S., A diffuse-interface model for axisymmetric immiscible two-phase flow, Appl. Math. Comput., 160 (2005), 589–606.Google Scholar
[58]Kim, J. S., A numerical method for the Cahn-Hilliard equation with a variable mobility, Commun. Nonlinear Sci. Numer. Simul., 12 (2007), 1560–1571.Google Scholar
[59]Kim, J. S., Phase field computations for ternary fluid flows, Comput. Meth. Appl. Mech. Eng., 196 (2007), 4779–4788.Google Scholar
[60]Kim, J. S., A generalized continuous surface tension force formulation for phase-field models for immiscible multi-component fluid flows, Comput. Meth. Appl. Mech. Eng., 198 (2009), 3105–3112.Google Scholar
[61]Kim, J. S. and Bae, H.-O., An unconditionally gradient stable adaptive mesh refinement for the Cahn-Hilliard equation, JKPS, 53(2) (2008), 2008–672.Google Scholar
[62]Kim, J. S. and Kang, K., A numerical method for the ternary Cahn-Hilliard system with a degenerate mobility, Appl. Numer. Math., 59 (2009), 1029–1042.CrossRefGoogle Scholar
[63]Kim, J. S., Kang, K. K. and Lowengrub, J. S., Conservative multigrid methods for Cahn-Hilliard fluids, J. Comput. Phys., 193 (2004), 511–543.Google Scholar
[64]Kim, J. S. and Lowengrub, J. S., Phase field modeling and simulation of three-phase flows, Int. Free Bound., 7 (2005), 435–466.CrossRefGoogle Scholar
[65]Kim, C.-H., Shin, S.-H., Lee, H. G. and Kim, J. S., A phase-field model for the pinchoff of liquid-liquid jets, JKPS, 55 (2009), 1451–1460.Google Scholar
[66]Kiwata, H., Instability of interfaces in phase-separating binary fluids at a finite Reynolds number, Phys. Fluids, 15 (2003), 2480–2485.Google Scholar
[67]Lee, H. G. and Kim, J. S., Accurate contact angle boundary conditions for the Cahn-Hilliard equations, Comput. Fluids, 44 (2011), 178–186.CrossRefGoogle Scholar
[68]Lee, H. G., Kim, K. M. and Kim, J. S., On the long time simulation of the Rayleigh-Taylor instability, Int. J. Numer. Meth. Eng., 85 (2011), 1633–1647.Google Scholar
[69]Lee, H. Y., Lowengrub, J. S. and Goodman, J., Modeling pinchoff and reconnection in a Hele-Shaw cell I: the models and their calibration, Phys. Fluids, 14(2) (2002), 2002–492.Google Scholar
[70]Lee, H. Y., Lowengrub, J. S. and Goodman, J., Modeling pinchoff and reconnection in a Hele-Shaw cell II: analysis and simulation in the nonlinear regime, Phys. Fluids, 14(2) (2002), 2002–514.Google Scholar
[71]Li, J. and Renardy, Y., Numerical study of flows of two immiscible liquids at low Reynolds number, SIAM Rev., 42(3) (2000), 2000–417.CrossRefGoogle Scholar
[72]Lighthill, J., Waves in Fluids, Cambridge University Press, Cambridge, 1978.Google Scholar
[73]Liu, C. and Shen, J., A phase field model for the mixture of two incompressible fluids and its approximation by a Fourier-spectral method, Phys. D, 179 (2003), 211–228.Google Scholar
[74]Lowengrub, J. S. and Truskinovsky, L., Quasi-incompressible Cahn-Hilliard fluids and topo-logical transitions, Proc. R. Soc. Lond. A, 454 (1998), 2617–2654.CrossRefGoogle Scholar
[75]Martin, D. F., Colella, P., Anghel, M. and Alexander, F. L., Adaptive mesh refinement for mul-tiscale nonequilibrium physics, Comput. Sci. Eng., 7 (2005), 24–31.Google Scholar
[76]Milosevic, I. N. and Longmire, E. K., Pinch-off modes and satellite formation in liquid-liquid jet systems, Int. J. Multiphase Flow, 28(11) (2002), 2002–1853.Google Scholar
[77]Novick-Cohen, A. and Segel, L. A., Nonlinear aspects of the Cahn-Hilliard equation, Phys. D, 10 (1984), 277–298.Google Scholar
[78]Osher, S. and Fedkiw, R. P., Level set methods: an overview and some recent results, J. Comput. Phys., 169 (2001), 463–502.CrossRefGoogle Scholar
[79]Osher, S. and Fedkiw, R. P., Level Set Methods and Dynamic Implicit Surfaces, Springer-Verlag, New York, 2002.Google Scholar
[80]Peskin, C. S., The immersed boundary method, Acta Num., 11 (2002), 1–39.Google Scholar
[81]Peskin, C. S. and McQueen, D. M., Modeling prosthetic heart valves for numerical analysis of blood flow in the heart, J. Comput. Phys., 37 (1980), 113–132.CrossRefGoogle Scholar
[82]Porter, D. A. and Easterling, K. E., Phase Transformations in Metals and Alloys, van Nostrand Reinhold, New York, 1993.Google Scholar
[83]Press, W. H., Teukolsky, S. A., Vetterling, W. T. and Flannery, B. P., Numerical Recipes in C, Cambridge University Press, New York, 1993.Google Scholar
[84]Rowlinson, J. S. and Widom, B., Molecular Theory of Capillarity, Dover Publications, New York, 2003.Google Scholar
[85]Sethian, J. A. and Smereka, P., Level set methods for fluid interfaces, Annu. Rev. Fluid Mech., 35 (2003), 341–372.Google Scholar
[86]Shen, J. and Yang, X., An efficient moving mesh spectral method for the phase-field model of two-phase flows, J. Comput. Phys., 228(8) (2009), 2009–2978.Google Scholar
[87]Shen, J. and Yang, X., Energy stable schemes for Cahn-Hilliard phase-field model of two-phase incompressible flows, Chin. Ann. Math., 31B(5) (2010), 2010–743.Google Scholar
[88]Smith, K. A., Solis, F. J. and Chopp, D. L., A projection method for motion of triple junctions by level sets, Int. Free Bound., 4 (2002), 263–276.Google Scholar
[89]Starovoitov, V. N., Model of the motion of a two-component liquid with allowance of capillary forces, J. Appl. Mech. Tech. Phys., 35 (1994), 891–897.Google Scholar
[90]Sun, Y. and Beckermann, C., Diffuse interface modeling of two-phase flow based on averaging: mass and momentum equations, Phys. D, 198 (2004), 281–308.Google Scholar
[91]Sussman, M., Almgren, A. S., Bell, J. B., Colella, P., Howell, L. H. and Welcome, M. L., An adaptive level set approach for incompressible two-phase flows, J. Comput. Phys., 148 (1999), 81–124.Google Scholar
[92]Sussman, M., Smereka, P. and Osher, S., A level set approach for computing solutions to in-compressible two-phase flow, J. Comput. Phys., 114 (1994), 146–159.Google Scholar
[93]Teigen, K. E., Song, P., Lowengrub, J. and Voigt, A., A diffuse-interface method for two-phase flows with soluble surfactants, J. Comput. Phys., 230 (2011), 375–393.Google Scholar
[94]Tomotika, S., On the instability of a cylindrical thread of a viscous liquid surrounded by another viscous fluid, Proc. Roy. Soc. A, 150 (1935), 322–327.Google Scholar
[95]Trottenberg, U., Oosterlee, C. and Schüller, A., MULTIGRID, Academic press, London, 2001.Google Scholar
[96]Udaykumar, H. S., Kan, H. C., Shyy, W. and Tran-Son-Tay, R., Multiphase dynamics in arbitrary geometries on fixed Cartesian grids, J. Comput. Phys., 137 (1997), 137–366.Google Scholar
[97]Uzgoren, E., Sin, J. and Shyy, W., Marker-based, 3-D adaptive cartesian grid method for multiphase flow around irregular geometries, Commun. Comput. Phys., 5(1) (2009), 2009–1.Google Scholar
[98]Verschueren, M., Vosse, F. N. Van De and Heijer, H. E. H., Diffuse-interface modelling of ther-mocapillary flow instabilities in a Hele-Shaw cell, J. Fluid Mech., 434 (2001), 153–166.Google Scholar
[99]Villanueva, W., Sjodahl, J., Stjernstrom, M., Roeraade, J. and Amberg, G., Microdroplet deposition under a liquid medium, Lanmuir, 23 (2007), 1171–1177.Google Scholar
[100]Vollmayr-Lee, B. P. and Rutenberg, A. D., Fast and accurate coarsening simulation with an unconditionally stable time step, Phys. Rev. E, 68 (2003), 066703.Google Scholar
[101]Wise, S., Kim, J. S. and Lowengrub, J. S., Solving the regularized, strongly anisotropic Cahn-Hilliard equation by an adaptive nonlinear multigrid method, J. Comput. Phys., 226 (2007), 414–446.Google Scholar
[102]Yang, X., Feng, J. J., Liu, C. and Shen, J., Numerical simulations of jet pinching-off and drop formation using an energetic variational phase-field method, J. Comput. Phys., 218 (2006), 417–428.Google Scholar
[103]Young, T., An essay on the cohesion of fluids, Trans. R. Soc. Lond., 95 (1805), 65–87.Google Scholar
[104]Yue, P. and Feng, J. J., Wall energy relaxation in the Cahn-Hilliard model for moving contact lines, Phys. Fluids, 23 (2011), 012106.CrossRefGoogle Scholar
[105]Yue, P., Feng, J. J., Liu, C. and Shen, J., A diffuse-interface method for simulating two-phase flows of complex fluids, J. Fluid Mech., 515 (2004), 293–317.Google Scholar
[106]Yue, P., Feng, J. J., Liu, C. and Shen, J., Diffuse-interface simulations of drop coalescence and retractio in viscoelastic fluids, J. Non-Newtonian Fluid Mech., 129 (2005), 163–176.Google Scholar
[107]Yue, P., Zhou, C. and Feng, J. J., Spontaneous shrinkage of drops and mass conservation in phase-field simulations, J. Comput. Phys., 223 (2007), 1–9.Google Scholar
[108]Zhang, S. and Wang, M., A nonconforming finite element method for the Cahn-Hilliard equation, J. Comput. Phys., 229 (2010), 7361–7372.Google Scholar
[109]Zhu, J., Chen, L. Q., Shen, J. and Tikare, V., Coarsening kinetics from a variable-mobility Cahn-Hilliard equation: application of a semi-implicit Fourier spectral method, Phys. Rev. E, 60 (1999), 3564–3572.Google Scholar